1 INTRODUCTION a - Duke Universitynikos/reprints/C-002-AdaptOptSPIE94.pdf · 1 INTRODUCTION Closed 1oop adaptive optics systems must compensate for time varying wave front distortions

Optimizing closed ioop adaptive optics performance using multiple control bandwidths

Brent L. Ellerbroek

Starfire Optical Range, US Air Force Phillips LaboratoryKirtland Air Force Base, New Mexico 87117

Charles Van Loan and Nikos P. Pitsianis

Department of Computer Science, Cornell UniversityIthaca, New York 14853

Robert J. Plemmons

Department of Mathematics and Computer Science, Wake Forest UniversityWinston-Salem, North Carolina 27109

ABSTRACT

The performance of a closed loop adaptive optics system may in principle be improved by selecting distinct andindependently optimized control bandwidths for separate components, or modes, of the wave front distortion profile.In this paper we outline a method for synthesizing and optimizing a multi-bandwidth adaptive optics control sys-tem from performance estimates previously derived for single-bandwidth control systems operating over a range ofbandwidths. Numerical results are presented for the case of an atmospheric turbulence profile consisting of a singletranslating phase screen with Kolmogorov statistics, a Shack-Hartmann wave front sensor with 8 subapertures acrossthe aperture of the telescope, and a continuous facesheet deformable mirror with actuators conjugate to the cornersof the wave front sensor subapertures. The use of multiple control bandwidths significantly relaxes the wave frontsensor noise level allowed for the adaptive optics system to operate near the performance limit imposed by fittingerror. Nearly all of this reduction is already achieved through the use of a control system utilizing only two distinctbandwidths, one of which is the zero bandwidth.

1 INTRODUCTION

Closed 1oop adaptive optics systems must compensate for time varying wave front distortions on the basis of noisysensor measurements. Time varying distortions are most accurately corrected by a system with a high controlbandwidth, but the noise in wave front sensor measurements is best rejected by reducing the control bandwidthand temporally smoothing the wave front correction to be applied. These trends lead to a performance tradeoffbetween the residual wave front distortions due to sensor noise and servo lag, and the optimum control bandwidthwhich minimizes the sum of these two effects will depend upon the wave front sensor noise level and the temporalcharacteristics of the wave front errors to be corrected. Most adaptive optics systems developed to date have addressedthis tradeoff using a common control bandwidth for all components of the wave front distortion profile 2. 3, but asystem employing several different control bandwidths for separate wave front components has recently been tested4 5 6, Because wave front sensor noise statistics and the temporal dynamics of the wave front distortion to becorrected vary as a function of spatial frequency 8,9, it should in principle be possible to improve adaptive opticsperformance by using this more sophisticated control approach. In this paper we describe a technique for evaluating

0819444964/94/$6.00 SPIE Vol. 2201 Adaptive Optics in Astronomy (1994)! 935

and optimizing the performance of adaptive optics systems using multiple control bandwidths. We also comparethe performance of the multi- and single bandwidth control approaches for a sample case involving atmosphericturbulence compensation.

Bandwidth optimization for a closed loop adaptive optics system using a single bandwidth can be readily accom-plished using standard results for the effect of sensor noise and servo lag on adaptive optics performance. Since thewave front errors due to these two effects may be treated as statistically independent, the combined mean-squarephase error a2(f) be minimized by adjusting the control bandwidth f takes the form

a2(f) =c(f)+ cr(f), (1)

where the terms o(f) and cr(f) are the mean-square phase errors due to sensor noise and servo lag. Explicit formulasfor these two quantities depend upon numerous parameters including wave front distortion statistics, wave front sensornoise statistics, the geometries for the wave front sensor subapertures and the deformable mirror actuators, the wavefront reconstruction algorithm, and the impulse response function of the adaptive optics control loop 10 The valueof f minimizing a2(f) may be determined numerically, and analytical solutions may be obtained when scaling lawapproximations are assumed for the functions o(f) and a(f) 12. 13, 14

The performance of a modal adaptive optics control system 15. 16may be improved further by applying the above

optimization process separately to each distinct component, or mode, of the wave front distortion profile. For asystem using an orthonormal basis of deformable mirror profiles as the set of modes to be controlled, Eq. (1) isreplaced by the formulas

2(f) (2)

cr�(fi) = (3)

where cT� is the mean-square residual phase error in mode number i, ft is the control bandwidth for this mode,and (a(f1) and (c(f1) are the statistically independent contributions of sensor noise and servo lag to the totalmean-square phase error in mode number i. Explicit formulas for these last two functions may be computed fromwave front distortion statistics and adaptive optics system parameters once an a priori basis of orthonormal controlmodes has been selected. The bandwidth ft minimizing Eq. (3) may then be determined numerically for each i.Zernike polynomials 17 are frequently used as the control mode basis for analytical calculations, although practicalapplications require that this basis set be adjusted to match the influence function characteristics of a particulardeformable mirror 16

One limitation for the above approach is that the choice of the control mode basis is external to the optimizationprocess. In this paper we describe a technique for improving the performance of an adaptive optics system bysimultaneously optimizing both the basis of wave front control modes and their associated control bandwidths.The inputs to this proceedure are a collection of positive definite n by n matrices , . . . , which have beenderived from the second-order statistics of single-bandwidth control systems for n different bandwidths sampling thebandwidth range of interest. The variable n denotes the number of deformable mirror degrees of freedom. Adaptiveoptics control modes and bandwidths are optimized by maximizing the functional f(U) => maxj{(UTMiU)2}over all n by n unitary matrices U. The columns of the optimal U define the orthogonal basis of control modesin terms of deformable mirror actuator commands. Mode number i is controlled at bandwidth number k preciselywhen the matrix Mc maximizes the quantity (UTMiU) over j = 1, . . . , n. The unitary U maximizing 1(U) iscomputed using an iterative algorithm in some ways analogous to the Jacobi method for diagonalizing a symmetricmatrix 18• The optimal U may also be determined explicitly from the eigenvectors of M1 in the special case where

= 2 and M2 0 because the second of the two control bandwidths is the zero bandwidth. We refer to this specialcase as reduced range single bandwidth control, since the range of the wave front reconstruction matrix is in thiscase a proper subspace of the vector space of deformable mirror actuator commands.

The remainder of the paper is organized as follows. Analytical results are outlined in Section 2, beginning witha review of single bandwidth adaptive optics systems in subsection 2.1. Subsection 2.2 evaluates the performance of

936 ISPIE Vol. 2201 Adaptive Optics in Astronomy (1994)

a multi-bandwidth control system for a prespecified set of control modes and bandwidths. Subsection 2.3 describeshow the functional 1(U) may be used to optimize the performance of a multi-bandwidth adaptive optics controlsystem, and subsection 2.4 considers the special case of reduced range single bandwidth control. Sample numericalresults are presented in Section 3, and Section 4 discusses the implications of these results for adaptive optics controlconcepts designed to adjust adaptive optics control bandwidths in real time to adjust to variations in sensor noiselevels and wave front distortion statistics.

2 ANALYTICAL RESULTS

Fig. 1 illustrates the closed loop adaptive optics control approaches to be investigated in this paper. Fig. la is a blockdiagram of the single control bandwidth approach. The top two-thirds of this figure represent the adaptive opticscontrol loop itself, and the bottom third describes the effect of the control system upon the optical performance ofthe telescope. Fig. lb illustrates an adaptive optics control loop employing multiple control bandwidths for separatemodes, or components, of the overall wave front distortion profile. The control loops and notation introduced inthese figures are described further in the paragraphs below. Formulas for evaluating and optimizing the opticalperformance of these systems are then developed in the following subsections.

The input disturbance to be nulled by the control system is the time varying rn-dimensional vector y (C) of wave-front slope sensor measurements. The vector y (t) includes the effect of wave-front sensor measurement noise, butnot the feedback applied to the sensor measurements by deformable mirror actuator adjustments. This second effectis included in the quantity y (t) — Cc (t), where c (t) is a n-dimensional vector comprised of the deformable mirroractuator position commands at time t, and C = ôy /Oc is the m by n Jacobian matrix of first-order changes inwave-front measurements y with respect to variations in the deformable mirror actuator command vector c .Eachcomponent of the vector c may correspond to a linear combination of commands to multiple physical actuators. It isconvenient to assume that the null space of the matrix C contains only the zero vector, so that all nonzero actuatorcommand vectors result in some effect on the wave front slope sensor measurement vector.

The single bandwidth control law used to determine deformable mirror actuator commands from the closed loopwave-front slope sensor measurement vector y —Cc is defined by the equations,

e = M(y —Cc), (4)

= ke. (5)

The n by m matrix M is referred to as the wave front reconstruction matrix, and the vector e is the estimate ofthe instantaneous closed ioop wave-front distortion represented as a sum of deformable mirror actuator adjustments.Eq. (5) relates the rate of change of the deformable mirror actuator command vector to the instantaneous wave frontdistortion estimate e . Note that all components of the command vector c are driven at the same bandwidth usingparallel single input, single output control laws. The particular form for this control law has been selected for thesake of simplicity and concreteness. Analogous results could be obtained below for any linear ordinary differentialequation with constant coefficients.

The wave front distortion profile to be corrected by the adaptive optics control loop is represented in Fig. la bythe notation 4(r ,0 ), and is a function of both the point r in the telescope aperture plane at which it is evaluatedand the direction 0 in the field-of-view of the telescope from which the phase profile has propagated. Since thephase profile 4(r , 0) and any modified profile of the form 4(r ,0)+ 1(0) will have identical effects upon the imagingperformance of the telescope, it is convenient to assume that the function cb(r, 0) has been adjusted by a functionof 0 to satisfy the condition

f dr WA(F )çb(r, 0) 0, (6)

where WA(r) is a {0, l}-valued function describing the clear aperture of the telescope. The function q5(r, 0) soadjusted will be referred to as a piston-removed phase profile. The adjustment icb(r, 0) applied to the phase profile

SPIE Vol. 2201 Adaptive Optics in Astronomy (1994) / 937

0psn loop imsuuzm.nt.

y

Qossd loop. oo..uoonmut.

y - Cc

PhOH

Opso loop—ofiloPi.ld-oIvlow-.rssd

phoss vsosncs

(a) Single control bandwidth

Qossd looprnsssartsy -Cc

Ssnstw

corrsccss

Actust

oplo loopzsosswsnsntsy

0—lo

(b) Multiple control bandwidths

Figure 1 : Adaptive optics system control loop dynamics.

Part (a) of the figure illustrates an adaptive optics control system in which all wave front modes are compensatedat the same servo bandwidth. Part (b) of the figure represents a system using different bandwidths and estimationalgorithms for distinct, orthogonal subspaces of wave front modes.

938 /SP!E Vol. 2201 Adaptive Optics in Astronomy (1994)

Ylsld-of-vi.w-ss.mgsd— vssisoss

by the adaptive optics system is assumed to be a linear function of the actuator command vector c and is describedby the equation

L/(r,O)=Jcjhi(r,O), (7)

where each function h(r , 0 ) is the adjustment to the phase profile resulting when a unit command is applied toactuator number i. This influence function will depend upon the field angle 0 when the deformable mirror is notconjugate to the aperture plane of the telescope. We will assume that overall piston has been removed from each ofthese influence functions, so that Eq. (6) applies with q5(r ,0 ) replaced with h(r , 0).

Eq. (7) will sometimes be abbreviated using operator notation in the form

içb=Hc. (8)

H is a linear operator by Eq. (7). The following analysis is simplified if we assume that deformable mirror actuatorcommands are expressed in terms of a basis set satisfying the condition

[Hc ,Hc '] = cTc , (9)

for any two actuator command vectors c and c '. This condition may always be obtained by orthogonalizing anyprespecified basis set of deformable mirror actuator commands.

The optical performance of the adaptive optics system is quantified using the expected mean-square value of theresidual phase distortion profile 4 — q5. This mean-square residual phase error is expressed in terms of an innerproduct [1 g] defined by the expression

I fdrfdOWA(r)WF(O)f(r,O)g(r,O)f,gj— fdrfdOWA(r)WF(O)where I and g are real-valued functions of r and 0 , and WF is a nonnegative function of 0 quantifying the relativeweight attached to different points in the field-of-view of the telescope. The expected mean-square residual phaseerror a2 is defined by the formula

a2 ([ _ zq5, 4 — LcA]) , (11)where the angle brackets denote ensemble averaging over the statistics of random sensor noise and phase distortionprofiles çb(r , 0 ). The details of how these averages are numerically calculated for specific wave front sensor anddeformable mirror configurations are described in previous work 19, 21, 22

The multiple bandwidth adaptive optics control system illustrated in Fig. lb replaces Eq.'s (4) and (5) by theestimation and control law

et = M(y —Cc), (12)

= kPe', (13)

c = (14)

The index i varies over the n different bandwidths employed by the adaptive optics control loop. Once again, theparticular loop compensation equation given in Eq. (13) is employed for the sake of simplicity and concreteness. Thebandwidth k' is used for deformable mirror adjustments within the range space of the orthogonal projection operatorP', and the total correction applied to the deformable mirror is the sum of these separately computed components.The projection operators P must satisfy the conditions

Pipi = P', (15)(pi)T = Pt, (16)PP = 0 for ij, (17)

= I. (18)


Eq.'s (15) and (16) are necessary and sufficient conditions for each matrix Pt to be an orthogonal projection operator18, 20

Eq.'s (17) and (18) imply that the range spaces of these projections decompose the space of all deformablemirror actuator commands into mutually orthogonal linear subspaces. The multiple bandwidth adaptive opticssystem illustrated in Fig. lb includes the previously described special case of modal control using a Zernike polynomialbasis for deformable mirror actuator commands 16

2.1 Single bandwidth systems

The steady-state solution to the single bandwidth control law described by Eq.'s (4) and (5) takes the form

C (t) =L dr k exp(—krMG)My (t —r). (19)

The deformable mirror actuator command vector c (t) is a highly nonlinear function of the reconstruction matrix M,and either optimizing or evaluating adaptive optics performance appears to be very difficult in this general case. Wetherefore introduce the constraint

MG = I, (20)

on the coefficients of the matrix M. Substituting Eq. (20) back into Eq. (19) gives the formula

c(t)=Ms(t), (21)

where the temporally filtered wave-front slope measurement vector is defined as

S (t) = L dT kexp(—kT)y (t — r). (22)

The deformable mirror actuator command vector c (t) is now a linear function of the coefficients of the wave-frontreconstruction matrix M. If Eq. (5) was replaced by another linear differential equation with scalar-valued, constantcoefficients, Eq. (22) for s (t) would become the corresponding steady-state solution with zero initial conditions.

It now follows that the expected mean-square residual phase error a2 is described by the expression

a2 2 _ tr(AMT) — tr(MAT) + tr(MSMT), (23)

where tr(V) denotes the trace of a square matrix V, and the quantities o, A, and S are defined by the formulas

cr = ([4,qS]), (24)A23 = (fr/',h1Js), (25)

= (sjs). (26)

Computational formulas for numerically evaluating these quantities for the case of Kolmogorov turbulence havebeen described previously 19 The formula for the expected mean-square residual phase distortion is quadratic inthe coefficients of the reconstruction matrix M, and the constraints imposed upon M by Eq. (20) are linear. Thevalue of M which minimizes the error cr2 may therefore be determined using Lagrange multipliers. The result is theexpression

M = [A5' + (I — AS_1G)(CTS_1G)_1GT5_i]. (27)

The minimum expected mean-square residual phase error for a single bandwidth adaptive optics control ioop at thebandwidth k may then be determined by substituting this value of M back in Eq. (23).

940 /SPIE Vol. 2201 Adaptive Optics in Astronomy (1994)

2.2 Evaluating multiple bandwidth systems

The first step in evaluating the performance of a multiple bandwidth adaptive optics control ioop is to derive anexpression for the actuator command vector c (t) equivalent to Eq. (21) in the single bandwidth case. We will onceagain assume a constraint of the form

MG = I, (28)

for each of the reconstruction matrices M, although the particular value of this matrix does not have to be derivedusing the results of the preceding subsection. It follows that the command vector c (t) is described by the expression

c(t) = PMs(t), (29)

where the temporally filtered wavefront sensor measurement vectors s(t) are defined by the formula

5 1(t) = i:° drk exp(—kr)y (t — r). (30)

The solution given by Eq. (29) is simply the sum of contributions from the individual wave-front reconstructionmatrices M on the range spaces of the projection operators Pt, and there is no crosseoupling between the separatereconstructors. This simplification would not have been obtained without Eq. (17).

A formula for the expected mean-square residual phase distortion of the multiple bandwidth adaptive opticssystem may now be obtained by substituting Eq.'s (8) and (29) into Eq. (11). The result is the expression

a2 2 _ tr(P1MP), (31)

where the matrices M' are defined by the formulas

M = Mi(Ai)T+ A1(Mi)T MiSn(Mi)T, (32)

Ak ([,h3J4) , (33)

s;,;., = (34)

We note for later use that the matrix M is symmetric. It follows from the definitions of the matrices A and S"that M satisfies the relationship

Mk ([h,, 4][hk, 4)]) _ ([hi, 4 — Ms'][hk, 4) _ Ms ']) . (35)

The matrix M describes the change to the second-order statistics ofthe correctable component ofthe phase distortionprofile resulting when the reconstruction matrix M and the control bandwidth k are employed in a single bandwidthadaptive optics control loop. Eq. (31) then illustrates the relationship between the overall performance of the multiplebandwidth control loop and the performance of the individual reconstruction matrices and bandwidths from whichit is constructed. There is no crosscoupling between different reconstruction algorithms in Eq. (31). This conditionwould not be obtained without the constraints on the projection operators P' imposed by Eq. (15) through (17).

2.3 Optimizing multiple bandwidth systems

Suppose now that single bandwidth reconstruction algorithms M have been optimized using Eq. (27) for a rangeof control bandwidths k, and that the corresponding matrices M defined by Eq. (32) have been computed. Weassume that the values of k' selected adequately sample the range of control bandwidths possibly of interest forthe given adaptive optics parameters and operating conditions. How do we determine the orthogonal projectionoperators P which will optimize the performance of the resulting multiple bandwidth adaptive optics control loop?

SPIE Vol. 2201 Adaptive Optics in Astronomy (1994)! 941

More precisely, we are interested in determining the value of the minimum expected mean-square residual phasedistortion a defined by the formula

= mm {a — tr(PMP) : (P' , . . . , P) Ep}

= a — max {tr(PiMiP) : (P', . . . , Pflc) E '} , (36)

where the set P is the collection of sequences of matrices (P', .. . , P') satisfying Eq. (15) through (18). Themaximum in Eq. (36) is taken over both the bandwidths at which modes are controlled and the control modesthemselves. This approach is more general than selecting optimal control bandwidths for each mode in a preselectedbasis set, such as Zernike polynomials. The value of a obtained will therefore be less than or equal to the bestperformance achievable with a prespecified basis set and the same range of allowable control bandwidths.

It may be shown that the minimized mean-square error a is described by the expression

= ci — max

{nmax(UTMU)ii : U unitarY} , (37)

where real-valued matrix U is unitary precisely when UTU = UUT J The columns of the unitary matrix Uassociated with this minimum become the control modes for the optimal control algorithm. Mode number j iscontrolled using bandwidth k precisely when (U'M1U) = maxk(UTMkU)jj. We have not found closed-formsolutions for o and the associated unitary matrix U in the general case. The objective function is neither a linearor differentiable function of the coefficients of U, and the collection of unitary matrices is not a convex set. Ourapproach to approximating a for sample numerical problems uses a pairwise, iterative optimization proceeduresomewhat analogous to the Jacobi method for diagonalizing a symmetric matrix.

2.4 Reduced range single bandwidth systems

The simplest possible multiple bandwidth control system employs one nonzero and one zero control bandwidth. Inthe present notation, this system is described by the parameters

n = 2, (38)k' = k, (39)k2 = 0. (40)

With k2 = 0, it follows quickly from the definitions given in Eq. (30), (33), (34), and (32) that

= 0, (41)A2 = 0, (42)S22 = 0, (43)M2 = 0, (44)

so that Eq. (37) for the minimum expected mean-square residual phase distortion becomes

= — max{max[(UTMU), 01: UunitarY}.

(45)

It may be shown that this formula simplifies to the closed-form solution

= — rnax(d, 0). (46)


where the quantities d3 are the eigenvalues of the symmetric matrix M. The control modes associated with thisminimum are just the eigenvectors of M, and a given mode is corrected at nonzero bandwidth precisely when theassociated eigenvalue is positive. This expression recovers an earlier result for the optimal range space of a wavefront reconstruction algorithm for systems employing a single nonzero control bandwidth 19

3 NUMERICAL RESULTS

In this section we apply the preceding analytical results to a sample case of atmospheric turbulence compensation.The atmospheric turbulence profile considered consists of a single thin phase screen with Kolmogorov statistics in theaperture plane of the telescope. The effects of a finite outer scale or a nonzero inner scale have not been included. Thetemporal dynamics of the phase screen are given by the Taylor (frozen flow) hypothesis, with a known windspeed vand a random, unknown wind direction. Under these conditions, the Greeenwood frequency fg 14 may be computedusing the expression

19 = O.423v/ro, (47)

where r0 is the turbulence-induced effective coherence diameter of the phase screen.

The adaptive optics system considered consists of a Shack-Hartmann wave front slope sensor and a continuousfacesheet deformable mirror with a linear spline actuator influence functiOn. The dimensionality of the system isgiven by the ratio D/d = 8, where D is the diameter of the circular, unobscured telescope aperture and d is the widthof an individual wave front sensor subaperture. This value yields a total of 88 fully illuminated and 44 partiallyilluminated Shack-Hartmann subapertures. Deformable mirror actuators are located conjugate to the corners of thesubapertures in the so—called Fried geometry . This yields 113 actuators within the telescope clear aperture, andan additional 44 edge acuators outside of but coupling into the clear aperture.

The WFS temporal sampling rate is assumed to be sufficiently large to support the highest control bandwidthof interest. Measurement noise has been treated as uncorrelated, temporally white for each individual subaperture,and uncorrelated between separate subapertures. Wave front sensor measurement noise is parameterized by thequantity a, defined as the mean-square phase difference measurement error for a fully illuminated subapertureand a sensor sampling rate equal to ten times the Greenwood frequency. Mean-square sensor noise for partiallyilluminated subapertures and other sampling rates is assumed to vary linearly with the sampling rate and inverselywith the area of the subaperture. These dependencies imply that detector noise can be neglected as a source of wavefront sensor measurement error.

Given the above assumptions, the performance of a single-bandwidth adaptive optics system using an optimizedwave front reconstruction algorithm is fully determined by the size of the aperture (D/d), the strength of theturbulence (D/ro), the Greenwood frequency f of the turbulence, the control loop bandwidth f = k/(2ir), andthe wave front sensor noise variance cr at the nominal sampling rate of 1Of. It may be shown that the residualmean-square phase error i2 for an optimized single-bandwidth adaptive optics system satisfies a scaling law of theform

a2 _ 2('f9O 0.7148

(d/ro)5/3 d ' Id '(d/ro)4/3

(

The normalized mean-square phase error e2 is a function of three quantities which may be interpreted as a normalizedaperture diameter, a normalized servo lag, and a normalized wave front sensor noise level. When applying this scalinglaw it is important to note that due to Eq. (47), both of the variables f and a depend upon the value of ro.

Fig. 2 plots the normalized mean-square residual phase error e2 as a function of the normalized servo lag and wavefront sensor noise level for a single-bandwidth adaptive optics system using an optimized wave front reconstructor.The best results are naturally obtained for the case of no wave front sensor noise, and in this case the residual meansquare phase error decreases monotonically with decreasing servo lag towards an asymptote imposed by the effectof fitting error. For each of the four nonzero noise levels considered there is an optimum control bandwidth whichminimizes the combined effect of servo lag and sensor noise. The curves for the residual phase variance do not follow


V

. 10

: 1

N

E0z 0.1

0.1

Figure 2: Performance of an adaptive optics control system using a single control bandwidth as a function ofnormalized noise level and servo lag.

These results were obtained using the results described in Section 2.1. They assume D/d= 8 and an unknownand random direction for the wind. The normalized residual phase variance is 2/(d/ro)5"3, the normalized servolag is (f9ro)/(fd), and the normalized RMS wavefront sensor noise is defined as cr/[2ir(d/ro)4"3].

the idealized (f9/f)5/3 dependance for large values of (fg/f) primarily because the telescope aperture diameter is. 23finite

Fig. 3 plots comparable results for the performance of an adaptive optics system using reduced range singlebandwidth control. These results were computed using Eq. (46) of Section 2.4. Simply not attempting to controlthose modes which are poorly sensed for a given level of wave front sensor noise and degree of servo lag yields asignificant reduction in the residual mean-square phase error. For a fixed bandwidth, the RMS wave front sensornoise level permitted to achieve a specified mean-square residual phase error is consistantly at least doubled throughthe use of reduced range single bandwidth control.

Table 1 lists corresponding results for a multi-bandwidth control system optimized over a range of control band-widths yielding normalized servo lags from 0.25 to 10. These results depend upon the normalized wave front sensornoise level but not the normalized servo lag, since the multi-bandwidth control algorithm selects an optimal controlbandwidth for each separate wave front control mode.

Fig. 4 compares the performance of the three different control options as a function of normalized noise level. Thecurve for the multi-bandwidth control system is taken directly from Table 1 . The curves for the single bandwidthand reduced range single bandwidth systems are derived from the minima of the curves plotted in Fig.'s 2 and 3,so that the single bandwidth used for this comparison is assumed to be optimized as a function of the normalizedwave front sensor noise level. The multi-bandwidth and reduced range single bandwidth approaches provide nearlyidentical performance improvements over the single bandwidth control system. The significant of this improvementdepends upon which axis of Fig. 4 is considered as the independent variable. The reduction in the mean-squarephase error for a fixed sensor noise level is never greater than a factor of two, but the relative change in the RMSsensor noise level permitted for a given residual mean-square phase error can be a considerably larger factor. Sincethe RMS wave front sensor noise level varies inversely with the square root of the number of photodetection effectsfor the case of photon noise, the corresponding reduction in the required wave front sensing beacon intensity is equal

944 /SPIE Vol. 2201 Adaptive Optics in Astronomy (1994)

Normalized RMSWFS Noise

1 10Normalized Servo Lag

0

N

E1.40z

I

I

C, I 0.002ir(d/ro)4/u(d/ro)513

0.316

0.05

0.369

0.10

0.449

0.15

0.6360.20 i

1.069

Table 1: Performance of a multi-bandwidth adaptive optics control system as a function of wave front sensor noisewith D/d = 8 and a random wind direction.

is the RMS wave front sensor phase difference measurement error in radians for a fully illuminated sub-aperture and a sampling rate equal to ten times the Greenwood frequency, d is the subaperture width, ro is theturbulence induced coherence diameter, and o2 is the residual mean-square phase error for the multi-bandwidthadaptive optics control system computed using the methods described in sections 2.2 and 2.3.


Normalized RMSWFS Noise

0.00— —0.05— — —0.10

1 0.200.40

0.10.1

Figure 3: Performance of an adaptive optics control system using reduced range single bandwidth control as afunction of normalized noise level and servo lag.

These results correspond to those illustrated in Fig. 2, except that they assume the use of a reduced range singlebandwidth control system as described in Section 2.4.

1 10Normalized Servo Lag

.l 1.2

0

G) 0.4

0—z

Figure 4: Performance comparison between single bandwidth control, reduced range single bandwidth control, andmulti-bandwidth control for D/d = 8 and a random wind direction.

The multi-bandwidth control results are taken from Table 1. The single bandwidth and reduced range singlebandwidth results are the minima of the curves plotted in Fig.'s 2 and 3. The definitions for normalized residualphase variance and normalized RMS WFS noise are as in Fig. 2.

to the square of this ratio.

The results in Fig. 's 2 through 4 and Table 1 all assume that D/d =8 and that the direction of the wind israndom and unknown. Similar results have been obtained for normalized aperture diameters with DId =12 andDid = 16, and also with a fixed and known wind direction. In none of the cases considered were the results formulti-bandwidth control and reduced range single bandwidth control significantly different.

4 DISCUSSION

This paper has described a technique for improving the performance of a modal adaptive optics control system bysimultaneously optimizing both the basis of wave front control modes and their associated control bandwidths. Aspecial case of this technique yields a control system employing only a single bandwidth on a reduced range of thepossible deformable mirror degrees of freedom. The two approaches yield nearly identical performance improvementsrelative to an adaptive optics control system using a single control bandwidth for all observable deformable mirrordegrees of freedom with an optimized wave front reconstruction algorithm. It may prove simpler in practice to adjusta single nonzero control bandwidth in real time than to simultaneously optimize an entire set of control bandwidths.

5 ACKNOWLEDGEMENTS

Brent L. Ellerbroek and Robert J. Plemmons acknowledge support from the Air Force Office of Scientific Research.


Solid: Single bandwidthDa8hed: Reduced range

single bandwidthDotted: Multiple bandwidths

0.1Normalized RMS WFS Noise

6 REFERENCES

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[2] F. Roddier, M. Northcott, and J. E. Graves, "A simple low-order adaptive optics system for near-infraredapplications," Pub. Astr. Soc. Pac. 103, 131—149 (1991).

[3] R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M.Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle,and R. A. Cleis, "Two generations of laser guide star adaptive optics experiments at the Starfire Optical Range,"J. Opt. Soc. Am. A 11, 310—325 (1994).

[4] G. Rousset et. al., "First diffraction limited astronomical images with adaptive optics," Astron. and Astrophys.230, L29—L32 (1990).

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1 INTRODUCTION a - Duke Universitynikos/reprints/C-002-AdaptOptSPIE94.pdf · 1 INTRODUCTION Closed 1oop adaptive optics systems must compensate for time varying wave front distortions